📚 Pythagoras' Theorem: Calculating Missing Sides in Right-Angled Triangles | 勾股定理：计算直角三角形缺失边长
In KS3 Cambridge Mathematics, one of the most powerful tools you will learn is Pythagoras' theorem. This fundamental principle reveals a precise relationship among the three sides of any right-angled triangle, allowing you to work out the length of one side when you know the other two. It is more than just a formula to memorise — it is a bridge between geometry and algebra that appears in construction, map-reading, engineering and even computer graphics.
在KS3剑桥数学课程中，你将学到的最强有力的工具之一就是勾股定理。这条基本原理揭示了任何直角三角形三条边之间的精确关系，让你在已知两边长度时能求出第三边的长度。它不仅是一个需要记住的公式，更是几何与代数之间的桥梁，广泛应用于建筑、地图判读、工程甚至计算机图形学中。
1. What is Pythagoras' Theorem? | 什么是勾股定理？
Pythagoras' nd the hypotenuse as c, the relationship is expressed by the famous equation c² = a² + b².
勾股定理指出，在任何直角三角形中，斜边（直角所对的边）长度的平方等于另外两边长度的平方和。如果我们把两条较短的边（通常称为直角边）标记为 a 和 b，斜边标记为 c，那么这种关系可以用著名的等式 c² = a² + b² 来表示。
c² = a² + b²
This equation only works for right-angled triangles. It tells us that the area of the square built on the hypotenuse is exactly the same as the combined area of the squares built on the other two sides.
这个等式仅适用于直角三角形。它告诉我们，以斜边为边长的正方形的面积，恰好等于以两条直角边为边长的两个正方形的面积之和。
2. Identifying the Sides of a Right-Angled Triangle | 识别直角三角形的边
Before applying the theorem, you must confidently identify each side. The hypotenuse is always the side opposite the right angle and is the longest side. The other two sides, often referred to as the adjacent and opposite (relative to a chosen acute angle), are the legs that form the right angle. In Pythagoras' theorem, it doesn't matter which leg you label a and which you label b, as long as c refers to the hypotenuse.
在应用定理之前，你必须能够准确地认出每一条边。斜边总是直角所对的边，也是三角形中最长的边。另外两条边常常被称为邻边和对边（相对于选定的锐角），它们是构成直角的两条直角边。在使用勾股定理时，你将哪条直角边标为 a、哪条标为 b 并不重要，只要 c 代表斜边即可。
he legs. If you are given the lengths of AB and BC, you can find AC by using c² = a² + b².
例如，在直角位于角 B 的三角形中，边 AC 就是斜边。边 AB 和 BC 是直角边。如果已知 AB 和 BC 的长度，你就可以用 c² = a² + b² 求出 AC。
3. The Formula and How to Use It | 公式及其使用方法
The basic steps for using Pythagoras' itute the known lengths and solve for the unknown side. Finally, take the square root if you are finding the hypotenuse, or rearrange and then take the square root if you are finding one of the shorter sides.
使用勾股定理的基本步骤很直接。首先，确认三角形包含一个直角。然后，标出各边，a 和 b 为直角边，c 为斜边。接着写下公式 c² = a² + b²。代入已知长度，求出未知边。最后，如果求的是斜边，直接取平方根；如果求的是某条直角边，则需先移项再取平方根。
square root. This is usually indicated by the √ symbol. For example, if c² = 25, then c = √25 = 5 cm.
我们必须始终牢记长度是正值，因此只取正的平方根。这通常用 √ 符号表示。例如，若 c² = 25，则 c = √25 = 5 cm。
4. Finding the Hypotenuse | 计算斜边
. Therefore c² = 100, and taking the square root gives c = 10 cm. The hypotenuse is 10 cm.
当你已知两条直角边而需要求斜边时，直接代入即可。假设一个直角三角形的两条直角边长分别是 6 cm 和 8 cm。写出等式：c² = 6² + 8²。计算平方：6² = 36，8² = 64。求和：36 + 64 = 100。因此 c² = 100，取平方根得到 c = 10 cm。斜边长为 10 cm。
This 6‑8‑10 combination is an example of a scaled Pythagorean triple (3‑4‑5 multiplied by 2). Recognising such triples can save time, but you should still show the working in exams.
这个 6‑8‑10 组合是勾股数（3‑4‑5 乘以 2）的一个缩放例子。识别这类勾股数可以节省时间，但在考试中你仍应展示计算过程。
5. Finding a Shorter Side | 计算较短直角边
= 25 + b². Subtract 25 from both sides: b² = 144. Taking the square root yields b = 12 m.
如果已知斜边和一条直角边，可以通过变形式求缺失的直角边。例如，一个直角三角形的斜边长为 13 m，其中一条直角边长为 5 m，设未知直角边为 b。那么有 13² = 5² + b²。即 169 = 25 + b²。两边减去 25 得 b² = 144。取平方根得到 b = 12 m。
mmon mistake is to write a² = c² + b², which would make the leg longer than the hypotenuse — an impossibility.
在求直角边时，务必记得使用减法而不是加法。常见的错误是写成 a² = c² + b²，这会让直角边长于斜边，而这是不可能的。
6. Applying Pythagoras' Theorem to Word Problems | 勾股定理应用题
ight-angled triangles. For example, a ladder leaning against a vertical wall forms a right‑ m ladder rests with its foot 1.5 m away from the wall, how high up the wall does it reach? is the other leg. Using h² + 1.5² = 5² gives h² + 2.25 = 25, so h² = 22.75 and h ≈ 4.77 m.
许多实际情景都可以用直角三角形来建模。例如，一把梯子斜靠在竖直墙壁上，会与地面和墙壁构成一个直角三角形。如果一把 5 m 长的梯子，底端离墙 1.5 m，那么它能到达墙多高的位置？这里梯子是斜边（5 m），离墙的距离是一条直角边（1.5 m），高度是另一条直角边。由 h² + 1.5² = 5² 得 h² + 2.25 = 25，因此 h² = 22.75，h ≈ 4.77 m。
Other classic problems include finding the diagonal of a rectangle, the shortest distance between two points on a coordinate grid, or checking whether a television screen fits a cabinet. Always draw a diagram and label the sides clearly.
其他经典问题包括求矩形的对角线长、坐标网格上两点之间的最短距离，或检查电视机屏幕能否放进电视柜。务必先画出简图并清晰标记各边。
7. Pythagorean Triples | 勾股数
mous triple is (3, 4, 5). Recognising common triples helps you spot answers quickly. Below is a table of some frequently used triples and their multiples.
勾股数是由三个正整数 a, b 和 c 组成的数组，满足 a² + b² = c²。最简单、最著名的勾股数是 (3, 4, 5)。认识常见的勾股数能帮助你快速识别答案。下面列出了一些常用勾股数及其倍数。
| a (leg) | b (leg) | c (hypotenuse) | Common multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6‑8‑10, 9‑12‑15, 15‑20‑25 |
| 5 | 12 | 13 | 10‑24‑26 |
| 7 | 24 | 25 | 14‑48‑50 |
| 8 | 15 | 17 | 16‑30‑34 |
plying (3,4,5) by 3 gives (9,12,15), which also satisfies 9² + 12² = 81 + 144 = 225 = 15².
任何勾股数的整数倍仍为勾股数。例如，将 (3,4,5) 乘以 3 得到 (9,12,15)，同样满足 9² + 12² = 81 + 144 = 225 = 15²。
8. The Converse of Pythagoras' Theorem | 勾股定理的逆定理
iangle has sides of lengths a, b and c satisfying a² + b² = c², then the triangle is right‑ ngle opposite the side c. This is extremely useful for testing whether a triangle is right‑angled without measuring angles.
该定理的逆定理指出，如果一个三角形的三边长 a, b 和 c 满足 a² + b² = c²，那么这个三角形就是直角三角形，且直角位于边 c 的对角。这对于不测量角度而检验一个三角形是否为直角三角形非常有用。
For example, to check if a triangle with sides 10 m, 24 m and 26 m is right‑ te 10² + 24² = 100 + 576 = 676, and 26² = 676. Since they are equal, the triangle is right‑angled, and the right angle lies opposite the 26 m side.
例如，要检查边长分别为 10 m, 24 m 和 26 m 的三角形是否为直角三角形，计算 10² + 24² = 100 + 576 = 676，而 26² = 676。两者相等，因此该三角形是直角三角形，且直角在 26 m 边的对角处。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
ation carefully: if c is the hypotenuse, then leg² = c² − known leg², not c² + known leg².
学生经常错误识别斜边，特别是当直角没有画在左下角时。务必首先确定直角位置：它所对的边就是斜边。另一个常见错误是在求较短边时用了加法而非减法。要仔细设置方程：若 c 为斜边，则直角边的平方为 c² − 已知直角边的平方，而不是 c² + 已知直角边的平方。
d of 13 loses marks. Finally, always include units in your final answer and check your rounding if the answer is not an integer.
最后忘记取平方根也是一个经典失误。在求得 c² = 169 后，写下 c = 169 而不是 13 就会丢分。最后，永远在最终答案中带上单位，并在答案不是整数时检查你的四舍五入。
10. Practice Questions and Tips | 练习题与技巧
Here are a few practice ideas to build fluency. Try finding the diagonal of a rectangle 12 cm by 5 cm. Then see how the theorem extends to three dimensions: the space diagonal of a cuboid with sides 3 cm, 4 cm and 12 cm can be found by applying Pythagoras twice. As a tip, always sketch a diagram — it helps you see which side is missing and avoid sign errors.
以下是几个提升熟练度的练习建议。试着求一个边长 12 cm 和 5 cm 的矩形的对角线长度。然后看看这一定理如何扩展到三维：求一个边长分别为 3 cm、4 cm 和 12 cm 的长方体的空间对角线，需要两次运用勾股定理。作为技巧，一定要画草图——这能帮助你看出缺失的是哪条边，并避免符号错误。
Remember, the key to mastering Pythagoras' theorem is not speed but understanding the relationship between the sides. Practise with a mixture of numerical exercises, word problems, and problems requiring the converse. With consistent effort, you'll find that what once seemed abstract becomes a reliable tool in your mathematical toolkit.
记住，掌握勾股定理的关键不在于速度，而在于理解边与边之间的关系。通过混合数值练习、应用题和需要逆定理的题目进行操练。持续努力之后，你会发现曾经看似抽象的内容将成为你数学工具箱中一件可靠的工具。
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